Is the promise to buy a phone for an employee in exchange for her transferring me to her manager or giving me his personal number considered a pledge that must be fulfilled, and is that considered a bribe?

Joking with young women is impermissible (haram) and leads to corruption. If there was an agreement between you to give her something in exchange for her service, then this must be fulfilled, based on the general evidence for the obligation to fulfill contracts. However, if giving it to her was merely a promise, then fulfilling it is recommended (mustahabb).